Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 15

Solution

Determine the set of points $z=x+iy$ that satisfy $|3z-2+i|=|3z+i|$.

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Solution

Let z=x+iyz=x+iy.

Given:

∣3z−2+i∣=∣3z+i∣|3z-2+i|=|3z+i|

Compute each term:

3z−2+i=3x+3iy−2+i=(3x−2)+i(3y+1)3z-2+i=3x+3iy-2+i=(3x-2)+i(3y+1) 3z+i=3x+3iy+i=3x+i(3y+1)3z+i=3x+3iy+i=3x+i(3y+1)

Square both sides:

(3x−2)2+(3y+1)2=(3x)2+(3y+1)2(3x-2)^2+(3y+1)^2=(3x)^2+(3y+1)^2

Cancel (3y+1)2(3y+1)^2:

(3x−2)2=(3x)2(3x-2)^2=(3x)^2

Expand:

9x2−12x+4=9x2⇒−12x+4=0⇒x=139x^2-12x+4=9x^2\Rightarrow -12x+4=0\Rightarrow x=\frac{1}{3}

So the locus is the vertical line x=13x=\frac{1}{3}:

x=13\boxed{x=\frac{1}{3}}

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